Linearithmic Clean-up for Vector-Symbolic Key-Value Memory with Kroneker Rotation Products

TL;DR

This paper introduces a novel codebook representation for Vector-Symbolic Architectures that enables linearithmic time clean-up, significantly improving scalability while maintaining comparable memory capacity.

Contribution

A new codebook method based on Kroneker products achieves efficient, scalable clean-up in VSAs with linearithmic complexity and minimal memory overhead.

Findings

Clean-up time complexity is reduced to O(N log N).

Memory space complexity remains at O(N).

Experiments show several orders of magnitude scalability improvement.

Abstract

A computational bottleneck in current Vector-Symbolic Architectures (VSAs) is the ``clean-up'' step, which decodes the noisy vectors retrieved from the architecture. Clean-up typically compares noisy vectors against a ``codebook'' of prototype vectors, incurring computational complexity that is quadratic or similar. We present a new codebook representation that supports efficient clean-up, based on Kroneker products of rotation-like matrices. The resulting clean-up time complexity is linearithmic, i.e. $\mathcal{O}(N\,\text{log}\,N)$, where $N$ is the vector dimension and also the number of vectors in the codebook. Clean-up space complexity is $\mathcal{O}(N)$. Furthermore, the codebook is not stored explicitly in computer memory: It can be represented in $\mathcal{O}(\text{log}\,N)$ space, and individual vectors in the codebook can be materialized in $\mathcal{O}(N)$ time and space. At the same time, asymptotic memory capacity remains comparable to standard approaches. Computer experiments confirm these results, demonstrating several orders of magnitude more scalability than baseline VSA techniques.